How can I crop and export parts of STL Objects?
Have you tried looking into the Boolean modifier?

If you object is a closed, watertight, manifold mesh you could use Boolean operations to break it into chunks of the desirable parts.

Just create a new closed watertight mesh object that as a shape or volume that engulfs or includes the part of the STL you want to keep and use it in a Boolean operation to intersect with the existing mesh and extract the part you want.

You can also try the Knife Project operator if a simpler cutting or slicing of the original geometry in one direction will suffice to obtain what you want.
You can use something as simple as a plane or a bezier curve to cut your geometry in the desired way.



Edit If your object is already composed of separate parts but all joined in a single object you can simply enter edit mode, select them all press P to separate them and select the option All loose parts to divide them into different objects.